Theorem 3eltr3g

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypotheses	3eltr3g.1	|- ( ph -> A e. B )
	3eltr3g.2	|- A = C
	3eltr3g.3	|- B = D
Assertion	3eltr3g	|- ( ph -> C e. D )

Proof

Step	Hyp	Ref	Expression
1		3eltr3g.1	|- ( ph -> A e. B )
2		3eltr3g.2	|- A = C
3		3eltr3g.3	|- B = D
4	2 1	eqeltrrid	|- ( ph -> C e. B )
5	4 3	eleqtrdi	|- ( ph -> C e. D )